Chapter 1Introduction

Abstract One characteristic that defines us, human beings, is the curiosity of theunknown. Since our birth, we have been trying to use any methods that humanbrains can comprehend to explore the nature: to mimic, to understand, and to utilizein a controlled and repeatable way. One of the most ancient means lies in thenature herself, experiments, leading to tremendous achievements from the creationof fire to the scissors of genes. Then comes mathematics, a new world we madeby numbers and symbols, where the nature is reproduced by laws and theoremsin an extremely simple, beautiful, and unprecedentedly accurate manner. With theexplosive development of digital sciences, computer was created. It provided usthe third way to investigate the nature, a digital world whose laws can be ruled byourselves with codes and algorithms to numerically mimic the real universe. In thischapter, we briefly review the history of tensor network algorithms and the relatedprogresses made recently. The organization of our lecture notes is also presented.

1.1 Numeric Renormalization Group in One Dimension

Numerical simulation is one of the most important approaches in science, inparticular for the complicated problems beyond the reach of analytical solutions.One distinguished example of the algorithms in physics as well as in chemistry isab initio principle calculation, which is based on density function theory (DFT) [1–3]. It provides a reliable solution to simulate a wide range of materials that can bedescribed by the mean-field theories and/or single-particle approximations. MonteCarlo method [4], named after a city famous of gambling in Monaco, is anotherexample that appeared in almost every corner of science. In contemporary physics,however, there are still many “hard nuts to crack.” Specifically in quantum physics,numerical simulation faces un-tackled issues for the systems with strong correla-tions, which might lead to exciting and exotic phenomena like high-temperaturesuperconductivity [5, 6] and fractional excitations [7].

Tensor network (TN) methods in the context of many-body quantum systemshave been developed recently. One could however identify some precursors of themin the seminal works of Kramers and Wannier [8, 9], Baxter [10, 11], Kelland [12],

© The Author(s) 2020S.-J. Ran et al., Tensor Network Contractions, Lecture Notes in Physics 964,https://doi.org/10.1007/978-3-030-34489-4_1

1

2 1 Introduction

Tsang [13], Nightingale and Blöte [11], and Derrida [14, 15], as found by Nishino[16–22]. Here we start their history from the Wilson numerical renormalizationgroup (NRG) [23]. The NRG aims at finding the ground state of a spin system.The idea of the NRG is to start from a small system whose Hamiltonian can beeasily diagonalized. The system is then projected on few low-energy states of theHamiltonian. A new system is then constructed by adding several spins and a newlow-energy effective Hamiltonian is obtained working only in the subspace spannedby the low-energy states of the previous step and the full Hilbert space of the newspins. In this way the low-energy effective Hamiltonian can be diagonalized againand its low-energy states can be used to construct a new restricted Hilbert space.The procedure is then iterated. The original NRG has been improved, for example,by combining it with the expansion theory [24–26]. As already shown in [23] theNRG successfully tackles the Kondo problem in one dimension [27], however, itsaccuracy is limited when applied to generic strongly correlated systems such asHeisenberg chains.

In the nineties, White and Noack were able to relate the poor NRG accuracywith the fact that it fails to consider properly the boundary conditions [28]. In 1992,White proposed the famous density matrix renormalization group (DMRG) that isas of today the most efficient and accurate algorithms for one-dimensional (1D)models [29, 30]. White used the largest eigenvectors of the reduced density matrixof a block as the states describing the relevant part of the low energy physics Hilbertspace. The reduced density matrix is obtained by explicitly constructing the groundstate of the system on a larger region. In other words, the space of one block isrenormalized by taking the rest of the system as an environment.

The simple idea of environment had revolutionary consequences in the RG-basedalgorithms. Important generalizations of DMRG were then developed, includingthe finite-temperature variants of matrix renormalization group [31–34], dynamicDMRG algorithms [35–38], and corner transfer matrix renormalization group byNishino and Okunishi [16].1

About 10 years later, TN was re-introduced in its simplest form of matrix productstates (MPS) [14, 15, 39–41] in the context of the theory of entanglement in quantummany-body systems; see, e.g., [42–45].2 In this context, the MPS encodes thecoefficients of the wave-functions in a product of matrices, and is thus defined asthe contraction of a one-dimensional TN. Each elementary tensor has three indexes:one physical index acting on the physical Hilbert space of the constituent, andtwo auxiliary indexes that will be contracted. The MPS structure is chosen sinceit represents the states whose entanglement only scales with the boundary of aregion rather than its volume, something called the “area law” of entanglement.Furthermore, an MPS gives only finite correlations, thus is well suited to represent

1We recommend a web page built by Tomotoshi Nishino, http://quattro.phys.sci.kobe-u.ac.jp/dmrg.html, where one exhaustively can find the progresses related to DMRG.2For the general theory of entanglement and its role in the physics of quantum many-body systems,see for instance [46–49].

1.2 Tensor Network States in Two Dimensions 3

the ground states of the gapped short-range Hamiltonians. The relation betweenthese two facts was evinced in seminal contributions [50–59] and led Verstraete andCirac to prove that MPS can provide faithful representations of the ground states of1D gapped local Hamiltonian [60].

These results together with the previous works that identified the outcome ofconverged DMRG simulations with an MPS description of the ground states [61]allowed to better understand the impressive performances of DMRG in terms ofthe correct scaling of entanglement of its underlying TN ansatz. The connectionbetween DMRG and MPS stands in the fact that the projector onto the effectiveHilbert space built along the DMRG iterations can be seen as an MPS. Thus, theMPS in DMRG can be understood as not only a 1D state ansatz, but also a TNrepresentation of the RG flows ([40, 61–65], as recently reviewed in [66]).

These results from the quantum information community fueled the search forbetter algorithms allowing to optimize variationally the MPS tensors in orderto target specific states [67]. In this broader scenario, DMRG can be seen asan alternating-least-square optimization method. Alternative methods include theimaginary-time evolution from an initial state encoded as in an MPS base of thetime-evolving block decimation (TEBD) [68–71] and time-dependent variationalprinciple of MPS [72]. Note that these two schemes can be generalized to simulatealso the short out-of-equilibrium evolution of a slightly entangled state. MPS hasbeen used beyond ground states, for example, in the context of finite-temperatureand low-energy excitations based on MPS or its transfer matrix [61, 73–77].

MPS has further been used to characterize state violating the area law ofentanglement, such as ground states of critical systems, and ground states ofHamiltonian with long-range interactions [56, 78–86].

The relevance of MPS goes far beyond their use as a numerical ansatz. Therehave been numerous analytical studies that have led to MPS exact solutions suchas the Affleck–Kennedy–Lieb–Tasaki (AKLT) state [87, 88], as well as its higher-spin/higher-dimensional generalizations [39, 44, 89–92]. MPS has been crucial inunderstanding the classification of topological phases in 1D [93]. Here we will nottalk about these important results, but we will focus on numerical applications eventhough the theory of MPS is still in full development and constantly new fieldsemerge such as the application of MPS to 1D quantum field theories [94].

1.2 Tensor Network States in Two Dimensions

The simulations of two-dimensional (2D) systems, where analytical solutions areextremely rare and mean-field approximations often fail to capture the long-rangefluctuations, are much more complicated and tricky. For numeric simulations, exactdiagonalization can only access small systems; quantum Monte Carlo (QMC)approaches are hindered by the notorious “negative sign” problem on frustratedspin models and fermionic models away from half-filling, causing an exponentialincrease of the computing time with the number of particles [95, 96].

4 1 Introduction

While very elegant and extremely powerful for 1D models, the 2D version ofDMRG [97–100] suffers several severe restrictions. The ground state obtained byDMRG is an MPS that is essentially a 1D state representation, satisfying the 1Darea law of entanglement entropy [52, 53, 55, 101]. However, due to the lack ofalternative approaches, 2D DMRG is still one of the most important 2D algorithms,producing a large number of astonishing works including discovering the numericevidence of quantum spin liquid [102–104] on kagomé lattice (see, e.g., [105–110]).

Besides directly using DMRG in 2D, another natural way is to extend the MPSrepresentation, leading to the tensor product state [111], or projected entangled pairstate (PEPS) [112, 113]. While an MPS is made up of tensors aligned in a 1D chain,a PEPS is formed by tensors located in a 2D lattice, forming a 2D TN. Thus, PEPScan be regarded as one type of 2D tensor network states (TNS). Note the work ofAffleck et al. [114] can be considered as a prototype of PEPS.

The network structure of the PEPS allows us to construct 2D states that strictlyfulfill the area law of entanglement entropy [115]. It indicates that PEPS canefficiently represent 2D gapped states, and even the critical and topological states,with only finite bond dimensions. Examples include resonating valence bond states[115–119] originally proposed by Anderson et al. for superconductivity [120–124],string-net states [125–127] proposed by Wen et al. for gapped topological orders[128–134], and so on.

The network structure makes PEPS so powerful that it can encode difficultcomputational problems including non-deterministic polynomial (NP) hard ones[115, 135, 136]. What is even more important for physics is that PEPS providesan efficient representation as a variational ansatz for calculating ground states of2D models. However, obeying the area law costs something else: the computationalcomplexity rises [115, 135, 137]. For instance, after having determined the groundstate (either by construction or variation), one usually wants to extract the physicalinformation by computing, e.g., energies, order parameters, or entanglement. For anMPS, most of the tasks are matrix manipulations and products which can be easilydone by classical computers. For PEPS, one needs to contract a TN stretching in a2D plain, unfortunately, most of which cannot be neither done exactly or nor evenefficiently. The reason for this complexity is what brings the physical advantage toPEPS: the network structure. Thus, algorithms to compute the TN contractions needto be developed.

Other than dealing with the PEPS, TN provides a general way to differentproblems where the cost functions are written as the contraction of a TN. A costfunction is usually a scalar function, whose maximal or minimal point gives thesolution of the targeted optimization problem. For example, the cost function of theground-state simulation can be the energy (e.g., [138, 139]); for finite-temperaturesimulations, it can be the partition function or free energy (e.g., [140, 141]); for thedimension reduction problems, it can be the truncation error or the distance beforeand after the reduction (e.g., [69, 142, 143]); for the supervised machine learningproblems, it can be the accuracy (e.g., [144]). TN can then be generally consideredas a specific mathematical structure of the parameters in the cost functions.

1.3 Tensor Renormalization Group and Tensor Network Algorithms 5

Before reaching the TN algorithms, there are a few more things worth men-tioning. MPS and PEPS are not the only TN representations in one or higherdimensions. As a generalization of PEPS, projected entangled simplex state wasproposed, where certain redundancy from the local entanglement is integrated toreach a better efficiency [145, 146]. Except for a chain or 2D lattice, TN can bedefined with some other geometries, such as trees or fractals. Tree TNS is oneexample with non-trivial properties and applications [39, 89, 147–158]. Anotherexample is multi-scale entanglement renormalization ansatz (MERA) proposed byVidal [159–167], which is a powerful tool especially for studying critical systems[168–173] and AdS/CFT theories ([174–180], see [181] for a general introductionof CFT). TN has also been applied to compute exotic properties of the physicalmodels on fractal lattices [182, 183].

The second thing concerns the fact that some TNs can indeed be contractedexactly. Tree TN is one example, since there is no loop of a tree graph. This mightbe the reason that a tree TNS can only have a finite correlation length [151], thuscannot efficiently access criticality in two dimensions. MERA modifies the tree ina brilliant way, so that the criticality can be accessed without giving up the exactlycontractible structure [164]. Some other exactly contractible examples have alsobeen found, where exact contractibility is not due to the geometry, but due to somealgebraic properties of the local tensors [184, 185].

Thirdly, TN can represent operators, usually dubbed as TN operators. Generallyspeaking, a TN state can be considered as a linear mapping from the physical Hilbertspace to a scalar given by the contraction of tensors. A TN operator is regardedas a mapping from the bra to the ket Hilbert space. Many algorithms explicitlyemploy the TN operator form, including the matrix product operator (MPO) forrepresenting 1D many-body operators and mixed states, and for simulating 1Dsystems in and out-of-equilibrium [186–196], tensor product operator (also calledprojected entangled pair operators) in for higher-systems [140, 141, 143, 197–206],and multiscale entangled renormalization ansatz [207–209].

1.3 Tensor Renormalization Group and Tensor NetworkAlgorithms

Since most of TNs cannot be contracted exactly (with #P-complete computationalcomplexity [136]), efficient algorithms are strongly desired. In 2007, Levin andNave generalized the NRG idea to TN and proposed tensor renormalization group(TRG) approach [142]. TRG consists of two main steps in each RG iteration:contraction and truncation. In the contraction step, the TN is deformed by singularvalue decomposition (SVD) of matrix in such a way that certain adjacent tensorscan be contracted without changing the geometry of the TN graph. This procedurereduces the number of tensors N to N/ν, with ν an integer that depends on the

6 1 Introduction

way of contracting. After reaching the fixed point, one tensor represents in factthe contraction of infinite number of original tensors, which can be seen as theapproximation of the whole TN.

After each contraction, the dimensions of local tensors increase exponentially,and then truncations are needed. To truncate in an optimized way, one shouldconsider the “environment,” a concept which appears in DMRG and is cruciallyimportant in TRG-based schemes to determine how optimal the truncations are.In the truncation step of Levin’s TRG, one only keeps the basis corresponding tothe χ -largest singular values from the SVD in the contraction step, with χ calleddimension cut-off. In other words, the environment of the truncation here is thetensor that is decomposed by SVD. Such a local environment only permits localoptimizations of the truncations, which hinders the accuracy of Levin’s TRG on thesystems with long-range fluctuations. Nevertheless, TRG is still one of the mostimportant and computationally cheap approaches for both classical (e.g., Ising andPotts models) and quantum (e.g., Heisenberg models) simulations in two and higherdimensions [184, 210–227]. It is worth mentioning that for 3D classical models, theaccuracy of the TRG algorithms has surpassed other methods [221, 225], such asQMC. Following the contraction-and-truncation idea, the further developments ofthe TN contraction algorithms concern mainly two aspects: more reasonable waysof contracting and more optimized ways of truncating.

While Levin’s TRG “coarse-grains” a TN in an exponential way (the numberof tensors decreases exponentially with the renormalization steps), Vidal’s TEBDscheme [68–71] implements the TN contraction with the help of MPS in a linearizedway [189]. Then, instead of using the singular values of local tensors, one uses theentanglement of the MPS to find the optimal truncation, meaning the environment isa (non-local) MPS, leading to a better precision than Levin’s TRG. In this case, theMPS at the fixed point is the dominant eigenstate of the transfer matrix of the TN.Another group of TRG algorithms, called corner transfer matrix renormalizationgroup (CTMRG) [228], are based on the corner transfer matrix idea originallyproposed by Baxter in 1978 [229], and developed by Nishina and Okunishi in 1996[16]. In CTMRG, the contraction reduces the number of tensors in a polynomialway and the environment can be considered as a finite MPS defined on the boundary.CTMRG has a compatible accuracy compared with TEBD.

With a certain way of contracting, there is still high flexibility of choosing theenvironment, i.e., the reference to optimize the truncations. For example, Levin’sTRG and its variants [142, 210–212, 214, 221], the truncations are optimized bylocal environments. The second renormalization group proposed by Xie et al. [221,230] employs TRG to consider the whole TN as the environments.

Besides the contractions of TNs, the concept of environment becomes moreimportant for the TNS update algorithms, where the central task is to optimize thetensors for minimizing the cost function. According to the environment, the TNSupdate algorithms are categorized as the simple [141, 143, 210, 221, 231, 232],cluster [141, 231, 233, 234], and full update [221, 228, 230, 235–240]. The simple

1.4 Organization of Lecture Notes 7

update uses local environment, hence has the highest efficiency but limited accuracy.The full update considers the whole TN as the environment, thus has a highaccuracy. Though with a better treatment of the environment, one drawback ofthe full update schemes is the expensive computational cost, which strongly limitsthe dimensions of the tensors one can keep. The cluster update is a compromisebetween simple and full update, where one considers a reasonable subsystem as theenvironment for a balance between the efficiency and precision.

It is worth mentioning that TN encoding schemes are found to bear closerelations to the techniques in multi-linear algebra (MLA) (also known as tensordecompositions or tensor algebra; see a review [241]). MLA was originally targetedon developing high-order generalization of the linear algebra (e.g., the higher-orderversion of singular value or eigenvalue decomposition [242–245]), and now hasbeen successfully used in a large number of fields, including data mining (e.g.,[246–250]), image processing (e.g., [251–254]), machine learning (e.g., [255]), andso on. The interesting connections between the fields of TN and MLA (for example,tensor-train decomposition [256] and matrix product state representation) open newparadigm for the interdisciplinary researches that cover a huge range in sciences.

1.4 Organization of Lecture Notes

Our lectures are organized as following. In Chap. 2, we will introduce the basicconcepts and definitions of tensor and TN states/operators, as well as their graphicrepresentations. Several frequently used architectures of TN states will be intro-duced, including matrix product state, tree TN state, and PEPS. Then the generalform of TN, the gauge degrees of freedom, and the relations to quantum entangle-ment will be discussed. Three special types of TNs that can be exactly contractedwill be exemplified in the end of this chapter.

In Chap. 3, the contraction algorithms for 2D TNs will be reviewed. We will startwith several physical problems that can be transformed to the 2D TN contractions,including the statistics of classical models, observation of TN states, and theground-state/finite-temperature simulations of 1D quantum models. Three paradigmalgorithms, namely TRG, TEBD, and CTMRG, will be presented. These algorithmswill be further discussed from the aspect of the exactly contractible TNs.

In Chap. 4, we will concentrate on the algorithms of PEPS for simulating theground states of 2D quantum lattice models. Two general schemes will be explained,which are the variational approaches and the imaginary-time evolution. Accordingto the choice of environment for updating the tensors, we will explain the simple,cluster, and full update algorithms. Particularly in the full update, the contractionalgorithms of 2D TNs presented in Chap. 3 will play a key role to compute the non-local environments.

8 1 Introduction

In Chap. 5, a special topic about the underlying relations between the TNmethods and the MLA will be given. We will start from the canonicalizationof MPS in one dimension, and then generalize to the super-orthogonalization ofPEPS in higher dimensions. The super-orthogonalization that gives the optimalapproximation of a tree PEPS in fact extends the Tucker decomposition from singletensor to tree TN. Then the relation between the contraction of tree TNs and therank-1 decomposition will be discussed, which further leads to the “zero-loop”approximation of the PEPS on the regular lattice. Finally, we will revisit the infiniteDMRG (iDMRG), infinite TEBD (iTEBD), and infinite CTMRG in a unified pictureindicated by the tensor ring decomposition, which is a higher-rank extension of therank-1 decomposition.

In Chap. 6, we will revisit the TN simulations of quantum lattice modelsfrom the ideas explained in Chap. 5. Such a perspective, dubbed as quantumentanglement simulation (QES), shows a unified picture for simulating one- andhigher-dimensional quantum models at both zero [234, 257] and finite [258]temperatures. The QES implies an efficient way of investigating infinite-size many-body systems by simulating few-body models with classical computers or artificialquantum platforms. In Chap. 7, a brief summary is given.

As TN makes a fundamental language and efficient tool to a huge range ofsubjects, which has been advancing in an extremely fast speed, we cannot coverall the related progresses in this review. We will concentrate on the algorithmsfor TN contractions and the closely related applications. The topics that are notdiscussed or are only briefly mentioned in this review include: the hybridizationof TN with other methods such as density functional theories and ab initiocalculations in quantum chemistry [259–268], the dynamic mean-field theory[269–278], and the expansion/perturbation theories [274, 279–284]; the TN algo-rithms that are less related to the contraction problems such as time-dependentvariational principle [72, 285], the variational TN state methods [76, 240, 286–291], and so on; the TN methods for interacting fermions [167, 266, 292–306],quantum field theories [307–313], topological states and exotic phenomena in many-body systems (e.g., [105, 106, 108, 110, 116–119, 125, 126, 306, 314–329]), theopen/dissipative systems [186, 190–192, 194, 330–334], quantum information andquantum computation [44, 335–344], machine learning [144, 345–360], and otherclassical computational problems [361–366]; the TN theories/algorithms with non-trivial statistics and symmetries [125–127, 303, 309, 314, 319, 367–380]; severallatest improvements of the TN algorithms for higher efficiency and accuracy[236, 239, 381–385].

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